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Quantum phase transitions: from Mott insulators to the cuprate superconductors. Colloquium article in Reviews of Modern Physics 75 , 913 (2003). - PowerPoint PPT Presentation
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Quantum phase transitions: from Mott insulators to the cuprate superconductors
Colloquium article in Reviews of Modern Physics 75, 913 (2003)
Leon Balents (UCSB) Eugene Demler (Harvard) Matthew Fisher (UCSB) Kwon Park (Maryland) Anatoli Polkovnikov (Harvard) T. Senthil (MIT) Ashvin Vishwanath (MIT) Matthias Vojta (Karlsruhe) Ying Zhang (Maryland)
Parent compound of the high temperature superconductors: 2 4La CuO
Cu
O
La
Band theory
k
k
Half-filled band of Cu 3d orbitals –ground state is predicted by band theory to be a metal.
However, La2CuO4 is a very good insulator
A Mott insulator
Parent compound of the high temperature superconductors: 2 4La CuO
Ground state has long-range spin density wave (Néel) order at wavevector K= ()
0 spin density wave order parameter:
; 1 on two sublatticesii i
S
S
spin operator with
angular momentum =1/2
ij i jij
i
H J S S
S
S
A Mott insulator
Parent compound of the high temperature superconductors: 2 4La CuO
Ground state has long-range spin density wave (Néel) order at wavevector K= ()
0 spin density wave order parameter:
; 1 on two sublatticesii i
S
S
spin operator with
angular momentum =1/2
ij i jij
i
H J S S
S
S
A Mott insulator
Parent compound of the high temperature superconductors: 2 4La CuO
Ground state has long-range spin density wave (Néel) order at wavevector K= ()
0 spin density wave order parameter:
; 1 on two sublatticesii i
S
S
spin operator with
angular momentum =1/2
ij i jij
i
H J S S
S
S
BCS
Doped state is a paramagnet with 0
and also a high temperature superconductor with
the BCS pairing order parameter 0
With increasing , there must be one or more
qua
.
nt
BCS
um phase transitions involving
( ) onset of a non-zero
( ) restoration of spin rotation invariance by a transition
from 0 to 0
i
ii
Introduce mobile carriers of density by substitutional doping of out-of-
plane ions e.g. 2 4La Sr CuO
Superconductivity in a doped Mott insulator
First study magnetic transition in Mott insulators………….
OutlineOutline
A. Magnetic quantum phase transitions in “dimerized” Mott insulators
Landau-Ginzburg-Wilson (LGW) theory
B. Mott insulators with spin S=1/2 per unit cellBerry phases, bond order, and the
breakdown of the LGW paradigm
C. Cuprate SuperconductorsCompeting orders and recent experiments
A. Magnetic quantum phase tranitions in “dimerized” Mott insulators:
Landau-Ginzburg-Wilson (LGW) theory:
Second-order phase transitions described by fluctuations of an order parameter associated with a broken symmetry
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.
S=1/2 spins on coupled dimers
jiij
ij SSJH
10
JJ
Coupled Dimer AntiferromagnetM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).
Weakly coupled dimersclose to 0
close to 0 Weakly coupled dimers
Paramagnetic ground state 0, 0iS
2
1
2
1
close to 0 Weakly coupled dimers
Excitation: S=1 triplon
close to 0 Weakly coupled dimers
2
1
Excitation: S=1 triplon
close to 0 Weakly coupled dimers
2
1
Excitation: S=1 triplon
close to 0 Weakly coupled dimers
2
1
Excitation: S=1 triplon
close to 0 Weakly coupled dimers
2
1
Excitation: S=1 triplon
close to 0 Weakly coupled dimers
2
1
Excitation: S=1 triplon
Energy dispersion away from antiferromagnetic wavevector
2 2 2 2
2x x y y
p
c p c p
spin gap
(exciton, spin collective mode)
TlCuCl3
N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).
“triplon”
/
For quasi-one-dimensional systems, the triplon linewidth takes
the exact universal value 1.20 at low TBk TBk Te
K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998)
This result is in good agreement with observations in CsNiCl3 (M. Kenzelmann, R. A. Cowley, W. J. L. Buyers, R. Coldea, M. Enderle, and D. F. McMorrow Phys. Rev. B 66, 174412 (2002)) and Y2NiBaO5 (G. Xu, C. Broholm, G. Aeppli, J. F. DiTusa, T.Ito, K. Oka, and H. Takagi, preprint).
Coupled Dimer Antiferromagnet
close to 1 Weakly dimerized square lattice
close to 1 Weakly dimerized square lattice
Excitations: 2 spin waves (magnons)
2 2 2 2p x x y yc p c p
Ground state has long-range spin density wave (Néel) order at wavevector K= ()
0
spin density wave order parameter: ; 1 on two sublatticesii i
S
S
TlCuCl3
J. Phys. Soc. Jpn 72, 1026 (2003)
1c
Néel state
T=0
Pressure in TlCuCl3
Quantum paramagnet
c = 0.52337(3) M.
Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)
The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,
Phys. Rev. Lett. 89, 077203 (2002))
0
0
1c
T=0
in cuprates
Quantum paramagnet
c = 0.52337(3) M.
Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)
Magnetic order as in La2CuO4 Electrons in charge-localized Cooper pairs
The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,
Phys. Rev. Lett. 89, 077203 (2002))
0
0
Néel state
22 22 2 22
1 1
2 4!x c
ud xd
cS
LGW theory for quantum criticality
write down an effective action
for the antiferromagnetic order parameter by expanding in powers
of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y
:
all s
ymmetries of the microscopic Hamiltonian
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
LGW theory for quantum criticality
write down an effective action
for the antiferromagnetic order parameter by expanding in powers
of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y
:
all s
ymmetries of the microscopic Hamiltonian
For , oscillations of about 0
constitute the excitationc
triplon
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
22 22 2 22
1 1
2 4!x c
ud xd
cS
B. Mott insulators with spin S=1/2 per unit cell:
Berry phases, bond order, and the breakdown of the LGW paradigm
Mott insulator with two S=1/2 spins per unit cell
Mott insulator with one S=1/2 spin per unit cell
Mott insulator with one S=1/2 spin per unit cell
Ground state has Neel order with 0
Mott insulator with one S=1/2 spin per unit cell
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
Mott insulator with one S=1/2 spin per unit cell
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
Mott insulator with one S=1/2 spin per unit cell
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,
and has 0, where is the
n bond order parameter
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,
and has 0, where is the
n bond order parameter
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,
and has 0, where is the
n bond order parameter
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,
and has 0, where is the
n bond order parameter
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Such a state breaks the symmetry of rotations by / 2 about lattice sites,
and has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,
which also has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,
which also has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,
which also has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,
which also has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Mott insulator with one S=1/2 spin per unit cell
bond bond
Another state breaking the symmetry of rotations by / 2 about lattice sites,
which also has 0, where is the
n bond order parameter
ar
ond
c
b
tan j ii
i jij
i S S e
r r
bond
Possible large paramagnetic gr Class ound state ( ) with 0A g
Resonating valence bonds
Resonance in benzene leads to a symmetric configuration of valence
bonds (F. Kekulé, L. Pauling)
bond
Different valence bond pairings
resonate with each other, leading
to
Cl
a reso
ass B
nating valen
paramagnet)
ce bond ,
( with 0
liquid
P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974); P.W. Anderson 1987
Such states are associated with non-collinear spin correlations, Z2 gauge theory, and topological order.N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991).
Excitations of the paramagnet with non-zero spin
bond 0; Class A
Excitations of the paramagnet with non-zero spin
bond 0; Class A
Excitations of the paramagnet with non-zero spin
bond 0; Class A
Excitations of the paramagnet with non-zero spin
bond 0; Class A
Excitations of the paramagnet with non-zero spin
bond 0; Class A
Excitations of the paramagnet with non-zero spin
bond 0; Class A
S=1/2 spinons, , are confined into a S=1
triplon, *~ z z
z
Excitations of the paramagnet with non-zero spin
bond 0; Class A
S=1/2 spinons, , are confined into a S=1
triplon, *~ z z
z
bond 0; Class B
Excitations of the paramagnet with non-zero spin
bond 0; Class A
S=1/2 spinons, , are confined into a S=1
triplon, *~ z z
z
bond 0; Class B
Excitations of the paramagnet with non-zero spin
bond 0; Class A
S=1/2 spinons, , are confined into a S=1
triplon, *~ z z
z
bond 0; Class B
Excitations of the paramagnet with non-zero spin
bond 0; Class A
S=1/2 spinons, , are confined into a S=1
triplon, *~ z z
z S=1/2 spinons can propagate
independently across the lattice
bond 0; Class B
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
iSAe
A
iSAe
A
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
Quantum theory for destruction of Neel order
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
, ,x y
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
2 aA
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
a
a
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
a
a
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
a
Change in choice of is like a “gauge transformation”
2 2a a a aA A
0
a
Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
a a
a
Recall = (0,0,1) in classical Neel state;
1 on two
square subl
2
attices ; a aS
a+
0
a
a a+ 0
oriented area of spherical triangle
formed by and an arbitrary reference , poi, nta hA alf
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
a
Change in choice of is like a “gauge transformation”
2 2a a a aA A
0
The area of the triangle is uncertain modulo 4and the action has to be invariant under 2a aA A
Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases
exp a aa
i A
Sum of Berry phases of all spins on the square lattice.
,
exp
with "current" of
charges 1 on sublattices
a aa
a
i J A
J
static
,
1a a
ag
2
,
11 expa a a a a a
a aa
Z d i Ag
Partition function on cubic lattice
LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
Quantum theory for destruction of Neel order
2
,
11 expa a a a a a
a aa
Z d i Ag
Partition function on cubic lattice
Modulus of weights in partition function: those of a classical ferromagnet at a “temperature” g
Berry phases lead to large cancellations between different
time histories need an effective action for at large aA g
Quantum theory for destruction of Neel order
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Small ground state has Neel order with 0
Large paramagnetic ground state with 0
g
g
2
2 2
,
with
This is compact QED in 3 spacetime dimensions with
static charges 1 on
1exp c
two sublat
tices.
o2
~
sa a a a aaa
Z dA A A i Ae
e g
Simplest large g effective action for the Aa
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).
bond (Class A parama
Analysis by a duality mapping shows that this theory
is in a phase with 0
The gauge theory is in a phase (spinons are confined
and only =1 triplons propagate
gnet).always
confining
S
Proliferation of monopoles in the presence of Berry phases.
).
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
Ordering by quantum fluctuations
g
Phase diagram of S=1/2 square lattice antiferromagnet
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
*
Neel order
~ 0z z
(associated with condensation of monopoles in ),A
or
bondBond order 0
1/ 2 spinons confined,
1 triplon excitations
S z
S
A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)
Bond order in a frustrated S=1/2 XY magnet
2 x x y yi j i j i j k l i j k l
ij ijkl
H J S S S S K S S S S S S S S
g=
First large scale (> 8000 spins) numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry
Mott insulators with spin S=1/2 per unit cell:
Berry phases, bond order, and the breakdown of the LGW paradigm
Order parameters/broken symmetry+
Emergent gauge excitations, fractionalization.
C. Cuprate superconductors:Competing orders and recent experiments
Magnetic Mott Insulator
Magnetic Superconductor
Paramagnetic Mott Insulator
Superconductor
BCS , 0 0 =
2 4La CuO
Quantum phasetransitions
High temperaturesuperconductor
BCS , 0 0 =
BCS , 0 0
BCS , 0 0
BCSMinimal LGW phase diagram with and
Magnetic Mott Insulator
Magnetic Superconductor
Paramagnetic Mott Insulator
Superconductor
BCS , 0 0 =
2 4La CuO
Quantum phasetransitions
BCS , 0 0 =
BCS , 0 0
BCS , 0 0
Spin density wave order ,K Spirals……Shraiman, Siggia Stripes……..Zaanen, Kivelson…..
BCSMinimal LGW phase diagram with and
High temperaturesuperconductor
Magnetic Mott Insulator
Paramagnetic Mott Insulator
BCS , 0 0 =
2 4La CuO
Quantum phasetransitions
BCS , 0 0 =
Magnetic Mott Insulator
Paramagnetic Mott Insulator
BCS , 0 0 =
2 4La CuO
Quantum phasetransitions
BCS , 0 0 =
g
Neel order
Bond order
La2CuO4
or
g
Neel order
Bond order
La2CuO4
or
Hole density
Large N limit of a theory with Sp(2N) symmetry: yields
existence of bond order and d-wave superconductivity
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991); M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Phys. Rev. B 66, 104505 (2002).
Localized holes
Magnetic, bondand super-conducting
order
g
Neel order
Bond order
La2CuO4
or
Hole density
Large N limit of a theory with Sp(2N) symmetry: yields
existence of bond order and d-wave superconductivity
S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991); M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999); M. Vojta, Phys. Rev. B 66, 104505 (2002).
Localized holes
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Neutron scattering measurements of La15/8Ba1/8CuO4 (Zurich oxide)
Scattering off spin density wave order with
cos
1 1 1 1 1 1 2 , and 2 ,
2 2 8 2 8 2
At higher energies, semiclassical theory predicts
that peaks lead to spin-wave ("light")
i iS N
Q r
Q =
cones.
xy
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Neutron scattering measurements of La15/8Ba1/8CuO4 (Zurich oxide)
Scattering off spin density wave order with
cos
1 1 1 1 1 1 2 , and 2 ,
2 2 8 2 8 2
At higher energies, semiclassical theory predicts
that peaks lead to spin-wave ("light")
i iS N
Q r
Q =
cones.
xy
A. T. Boothroyd, D. Prabhakaran, P. G. Freeman, S.J.S. Lister, M. Enderle, A. Hiess, and J. Kulda, Phys. Rev. B 67, 100407 (2003).
La5/3Sr1/3NiO4
1 1 1 1Ni has 1; 2 ,
2 6 2 6S
Q
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Neutron scattering measurements of La15/8Ba1/8CuO4 (Zurich oxide)
Scattering off spin density wave order with
cos
1 1 1 1 1 1 2 , and 2 ,
2 2 8 2 8 2
At higher energies, semiclassical theory predicts
that peaks lead to spin-wave ("light")
i iS N
Q r
Q =
cones.
xy
A. T. Boothroyd, D. Prabhakaran, P. G. Freeman, S.J.S. Lister, M. Enderle, A. Hiess, and J. Kulda, Phys. Rev. B 67, 100407 (2003).
La5/3Sr1/3NiO4
Spin waves: J=15 meV, J’=7.5meV
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Scattering off spin density wave order with
cos
1 1 1 1 1 1 2 , and 2 ,
2 2 8 2 8 2
At higher energies, semiclassical theory predicts
that peaks lead to spin-wave ("light")
i iS N
Q r
Q =
cones.
xy
A. T. Boothroyd, D. Prabhakaran, P. G. Freeman, S.J.S. Lister, M. Enderle, A. Hiess, and J. Kulda, Phys. Rev. B 67, 100407 (2003).
La5/3Sr1/3NiO4
Spin waves: J=15 meV, J’=7.5meV
Neutron scattering measurements of La15/8Ba1/8CuO4 (Zurich oxide)
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Observations in La15/8Ba1/8CuO4
are very different and do not obey spin-wave model.
Similar spectra are seen in most hole-doped cuprates.
xy
“Resonance peak”
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Red lines: triplon excitation of a 2 leg
ladder with exchange J=100 meV
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Red lines: triplon excitation of a 2 leg
ladder with exchange J=100 meV
J. M. Tranquada, H. Woo, T. G. Perring, H. Goka, G. D. Gu, G. Xu, M. Fujita, and K. Yamada, cond-mat/0401621
Red lines: triplon excitation of a 2 leg
ladder with exchange J=100 meV
J. M. Tranquada et al., cond-mat/0401621
xy
Spectrum of a two-leg ladder
Possible simple microscopic model of bond order
Spin waves
Triplons
“Resonance peak”
• M. Vojta and T. Ulbricht, cond-mat/0402377• G.S. Uhrig, K.P. Schmidt, and M. Grüninger, cond-mat/0402659• M. Vojta and S. Sachdev, unpublished.
J. M. Tranquada et al., cond-mat/0401621
xy
Bond operator (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) theory of coupled-ladder model,
M. Vojta and T. Ulbricht, cond-mat/0402377
J. M. Tranquada et al., cond-mat/0401621
xy
Numerical study of coupled ladder model, G.S. Uhrig, K.P. Schmidt, and M. Grüninger,
cond-mat/0402659
J. M. Tranquada et al., cond-mat/0401621
xy
LGW theory of magnetic criticality in the presence of static bond order, M. Vojta and S. Sachdev, to appear.
G.S. Uhrig, K.P. Schmidt, and M. Grüninger, cond-mat/0402659
Conclusions
I. Theory of quantum phase transitions between magnetically ordered and paramagnetic states of Mott insulators:
A. Dimerized Mott insulators: Landau-Ginzburg-Wilson theory of fluctuating magnetic order parameter.
B. S=1/2 square lattice: Berry phases induce bond order, and LGW theory breaks down. Critical theory is expressed in terms of emergent fractionalized modes, and the order parameters are secondary.
Conclusions
I. Theory of quantum phase transitions between magnetically ordered and paramagnetic states of Mott insulators:
A. Dimerized Mott insulators: Landau-Ginzburg-Wilson theory of fluctuating magnetic order parameter.
B. S=1/2 square lattice: Berry phases induce bond order, and LGW theory breaks down. Critical theory is expressed in terms of emergent fractionalized modes, and the order parameters are secondary.
Conclusions
II. Competing spin-density-wave/bond/superconducting orders in the hole-doped cuprates.
• Main features of spectrum of excitations in LBCO modeled by LGW theory of quantum critical fluctuations in the presence of static bond order across a wide energy range.
• Predicted magnetic field dependence of spin-density-wave order observed by neutron scattering in LSCO. E. Demler, S. Sachdev, and Y. Zhang, Phys.Rev. Lett. 87, 067202 (2001); B. Lake et al. Nature, 415, 299 (2002); B. Khaykhovich et al. Phys. Rev. B 66, 014528 (2002).
• Predicted pinned bond order in vortex halo consistent with STM observations in BSCCO. K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Phys. Rev. B 66, 094501 (2002); J.E. Hoffman et al. Science 295, 466 (2002).
• Energy dependence of LDOS modulations in BSCCO best modeled by modulations in bond variables. M. Vojta, Phys. Rev. B 66, 104505 (2002); D.
Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B 67, 094514 (2003); C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).
Conclusions
II. Competing spin-density-wave/bond/superconducting orders in the hole-doped cuprates.
• Main features of spectrum of excitations in LBCO modeled by LGW theory of quantum critical fluctuations in the presence of static bond order across a wide energy range.
• Predicted magnetic field dependence of spin-density-wave order observed by neutron scattering in LSCO. E. Demler, S. Sachdev, and Y. Zhang, Phys.Rev. Lett. 87, 067202 (2001); B. Lake et al. Nature, 415, 299 (2002); B. Khaykhovich et al. Phys. Rev. B 66, 014528 (2002).
• Predicted pinned bond order in vortex halo consistent with STM observations in BSCCO. K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Phys. Rev. B 66, 094501 (2002); J.E. Hoffman et al. Science 295, 466 (2002).
• Energy dependence of LDOS modulations in BSCCO best modeled by modulations in bond variables. M. Vojta, Phys. Rev. B 66, 104505 (2002); D.
Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B 67, 094514 (2003); C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).
Conclusions
III. Breakdown of LGW theory of quantum phase transitions with magnetic/bond/superconducting orders in doped Mott insulators ?
Conclusions
III. Breakdown of LGW theory of quantum phase transitions with magnetic/bond/superconducting orders in doped Mott insulators ?